New answers tagged boundary-conditions
1
Your code does not solve the BVP you posted. Here is the revised version that works well.
function bvp(N)
[D,x] = cheb(N); % Set up differentiation matrix
D2 = D^2;
% Insert boundary condition: u(-1) = 0, u(1) = 2
D2(1,:) = zeros(1,N+1);
D2(1,1) = 1;
D2(N,:) = zeros(1,N+1);
D2(N,N) = 1;
% right hand side
f = zeros(N+1,1);
f(1) = 2;
...
2
You cannot specify just a boundary condition on $\partial_x u$ at $x_L$.
Remember that in DG your solution is composed of piecewise discontinuous polynomials; at every interface you can have effectively any jump in $u$, which is separate from the slope of u to the left or right of the interface.
Unless $f(u,x)$ has any dependence on $\partial_x u$ (in which ...
3
The Lax-Friedrichs flux roughly approximates a situation where the propagation of information can occur in both directions. This is why on the right face at $x_{N+1}$ you need to also know $u^{N+1}_h$ in order to compute this numerical flux.
As an alternative, you could use an upwinding flux (either just at the boundaries or in your entire domain).
These ...
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